#253339

order	512
nilpotency class	3
isoclinism	isoclinic class of #253339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7)(3,11)(4,15)(8,24)(9,12)(10,22)(13,27)(14,29)(16,20)(17,18)(21,26)(23,31)(25,32)(28,30);(2,8,10,16)(3,11)(7,20,22,24)(9,25,26,31)(12,23,21,32)(13,27)(14,28)(29,30);(1,2)(3,12)(4,16)(5,10)(6,9)(7,11)(8,17)(13,21)(14,24)(15,25)(18,31)(19,26)(20,29)(22,27)(23,30)(28,32);(2,9)(3,13)(7,21)(8,25)(10,26)(11,27)(12,22)(14,29)(16,31)(20,32)(23,24)(28,30)

Polycyclic presentation:

2^9|b^2=e,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160535);